Question: $ C = \left[\begin{array}{rrr}-2 & -1 & 3 \\ 4 & 4 & -1\end{array}\right]$ $ E = \left[\begin{array}{rr}-1 & -1 \\ -1 & 0 \\ 0 & 5\end{array}\right]$ What is $ C E$ ?
Because $ C$ has dimensions $(2\times3)$ and $ E$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C E = \left[\begin{array}{rrr}{-2} & {-1} & {3} \\ {4} & {4} & {-1}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{-1} \\ {-1} & \color{#DF0030}{0} \\ {0} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-2}\cdot{-1}+{-1}\cdot{-1}+{3}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{-1}+{-1}\cdot{-1}+{3}\cdot{0} & ? \\ {4}\cdot{-1}+{4}\cdot{-1}+{-1}\cdot{0} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{-2}\cdot{-1}+{-1}\cdot{-1}+{3}\cdot{0} & {-2}\cdot\color{#DF0030}{-1}+{-1}\cdot\color{#DF0030}{0}+{3}\cdot\color{#DF0030}{5} \\ {4}\cdot{-1}+{4}\cdot{-1}+{-1}\cdot{0} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-2}\cdot{-1}+{-1}\cdot{-1}+{3}\cdot{0} & {-2}\cdot\color{#DF0030}{-1}+{-1}\cdot\color{#DF0030}{0}+{3}\cdot\color{#DF0030}{5} \\ {4}\cdot{-1}+{4}\cdot{-1}+{-1}\cdot{0} & {4}\cdot\color{#DF0030}{-1}+{4}\cdot\color{#DF0030}{0}+{-1}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}3 & 17 \\ -8 & -9\end{array}\right] $